2. THEORY AND RESULTS

Now we study the single-body dynamics of charged particle in a compound field: a DC electric field E _y = G (where G is a constant) and a laser (E _x = E _m sin (kz–ωt), B _y = B _m sin (kz−ωt)). Here, the laser propagates in vacuum and hence meets two relations: E _m = cB _m and ω = kc. Dimensionless 3D relativistic Newton equation set (RNEs) read

(1)$$d_s \left[{\Gamma d_s \displaystyle{y \over {\rm \lambda} }} \right]= - \displaystyle{{{\rm \omega} _G } \over {\rm \omega} }\comma \;$$

(2)$$d_s \left[{\Gamma d_s \displaystyle{z \over {\rm \lambda} }} \right]= - \displaystyle{{{\rm \omega} _B } \over {\rm \omega} }d_s \displaystyle{x \over {\rm \lambda} } \ast \sin{\rm \theta}\comma \;$$

(3)$$d_s \left[{\Gamma d_s \displaystyle{x \over {\rm \lambda} }} \right]= - \displaystyle{{{\rm \omega} _B } \over {\rm \omega} }\left[{1 - d_s \displaystyle{z \over {\rm \lambda} }} \right]\ \ast \sin{\rm \theta}\comma \;$$

where ${\rm \lambda } = 2{\rm \pi }{c \over {\rm \omega} }$, ${\rm \theta} = {{z - ct} \over {\rm \lambda}}\, \ast\, 2{\rm \pi }$, $s = {\rm \omega} t$, ${\rm \omega} _B = {{eB} \over {m_e }}$ and ${\rm \omega} _G = {{eG} \over {m_e c}}$. Eqs.(1), (3) will yield

(4)$$\Gamma d_s \displaystyle{y \over {\rm \lambda}} = - \displaystyle{{\rm \omega}_G \over {\rm \omega}}s\comma \;$$

$$\Gamma d_s \displaystyle{x \over {\rm \lambda}} + \displaystyle{{\rm \omega}_B \over {\rm \omega}} \ast \cos {\rm \theta} = {cons} \tan t = \displaystyle{{\rm \omega}_B \over {\rm \omega}}\comma \;$$

Here, the value of constant in Eq. (5) is determined by the initial condition (at t = 0, there are $d_s {x \over {\rm \lambda} } = 0$, Γ = 1 and θ = 0), and hence Eq.(5) can be re-written as

(6)$$\Gamma d_s \displaystyle{x \over {\rm \lambda} } = - \displaystyle{{{\rm \omega} _B } \over {\rm \omega} }\, \ast\, 2 \sin^2 \left({\displaystyle{{\rm \theta} \over 2}} \right).$$

Due to the definition $1/\Gamma = \sqrt {1 - \mathop {\left({d_s {x \over {\rm \lambda} }} \right)}\nolimits^2 - \mathop {\left({d_s {y \over {\rm \lambda} }} \right)}\nolimits^2 - \mathop {\left({d_s {z \over {\rm \lambda} }} \right)}\nolimits^2 } $, Eqs.(4), (6) will yield

(7)$$\mathop {\left({d_s \displaystyle{x \over {\rm \lambda} }} \right)}\nolimits^2 = \displaystyle{{\mathop {\left({ - {{{\rm \omega} _B } \over {\rm \omega} } \,\ast\, 2 \sin^2 \left({{{\rm \theta} \over 2}} \right)} \right)}\nolimits^2 } \over {\left[{1 + \mathop {\left({ - {{{\rm \omega} _B } \over {\rm \omega} } \,\ast\, 2 \sin^2 \left({{{\rm \theta} \over 2}} \right)} \right)}\nolimits^2 } \right]}} \,\ast\, \left[{1 - \mathop {\left({d_s \displaystyle{y \over {\rm \lambda} }} \right)}\nolimits^2 - \mathop {\left({d_s \displaystyle{z \over {\rm \lambda} }} \right)}\nolimits^2 } \right]\comma \;$$

$$\mathop {\left({d_s \displaystyle{y \over {\rm \lambda} }} \right)}\nolimits^2 = \displaystyle{{\mathop {\left({ - {{{\rm \omega} _G } \over {\rm \omega} }s} \right)}\nolimits^2 } \over {1 + \mathop {\left({ - {{{\rm \omega} _G } \over {\rm \omega} }s} \right)}\nolimits^2 }} \,\ast\, \left[{1 - \mathop {\left({d_s \displaystyle{x \over {\rm \lambda} }} \right)}\nolimits^2 - \mathop {\left({d_s \displaystyle{z \over {\rm \lambda} }} \right)}\nolimits^2 } \right]\comma \;$$

and hence

(9)$$\mathop {\left({d_s \displaystyle{y \over {\rm \lambda} }} \right)}\nolimits^2 = \displaystyle{{\mathop {\left({ - {{{\rm \omega} _G } \over {\rm \omega} }s} \right)}\nolimits^2 } \over {1 + \mathop {\left({ - {{{\rm \omega} _G } \over {\rm \omega} }s} \right)}\nolimits^2 + \mathop {\left({ - {{{\rm \omega} _B } \over {\rm \omega} } \,\ast\, 2 \sin^2 \left({{{\rm \theta} \over 2}} \right)} \right)}\nolimits^2 }} \,\ast\, \left[{1 - \mathop {\left({d_s \displaystyle{z \over {\rm \lambda} }} \right)}\nolimits^2 } \right]\comma \;$$

(10)$$\mathop {\left({d_s \displaystyle{x \over {\rm \lambda} }} \right)}\nolimits^2 = \displaystyle{{\mathop {\left({ - {{{\rm \omega} _B } \over {\rm \omega} } \,\ast\, 2 \sin^2 \left({{{\rm \theta} \over 2}} \right)} \right)}\nolimits^2 } \over {1 + \mathop {\left({ - {{{\rm \omega} _G } \over {\rm \omega} }s} \right)}\nolimits^2 + \mathop {\left({ - {{{\rm \omega} _B } \over {\rm \omega}} \,\ast\, 2\sin^2 \left({{{\rm \theta} \over 2}} \right)} \right)}\nolimits^2 }} \,\ast\, \left[{1 - \mathop {\left({d_s \displaystyle{z \over {\rm \lambda} }} \right)}\nolimits^2 } \right].$$

For convenience in expressing formulas, we introduce symbols

(11)$$\Delta = \sqrt {1 + \mathop {\left({ - \displaystyle{{{\rm \omega} _G } \over {\rm \omega} }s} \right)}\nolimits^2 + \mathop {\left({ - \displaystyle{{{\rm \omega} _B } \over {\rm \omega} } \,\ast\, 2\sin^2 \left({\displaystyle{{\rm \theta} \over 2}} \right)} \right)}\nolimits^2 }\semicolon \;$$

(12)$$M = \sqrt {1 - \mathop {\left({d_s \displaystyle{{\rm z} \over {\rm \lambda} }} \right)}\nolimits^2 } = \sqrt { - 2 \,\ast\, d_s {\rm \theta} - \mathop {\left({d_s {\rm \theta} } \right)}\nolimits^2 } \comma \;$$

and hence Eqs.(4), (9) imply

(13)$$\Gamma = \displaystyle{{\left({ - {\textstyle{{{\rm \omega} _G } \over {\rm \omega} }}s} \right)} \over {d_s {\textstyle{y \over {\rm \lambda} }}}} = \displaystyle{\Delta \over M}.$$

Thus, Eq.(2) is re-written as

(14)$$d_s \left[{\Delta \sqrt {\displaystyle{1 \over {M^2 }} - 1} } \right]= - \displaystyle{{{\rm \omega} _B } \over {\rm \omega} } \,\ast\, \displaystyle{{\left({ - {{{\rm \omega} _B } \over {\rm \omega}} \,\ast\, 2\sin^2 \left({{{\rm \theta} \over 2}} \right)} \right)} \over \Delta }{M} \,\ast \sin{\rm \theta}.$$

From definitions Eqs.(11), (12), we have

(15)$$d_s \Delta = \displaystyle{{\mathop {\left({ - {{{\rm \omega} _G } \over {\rm \omega} }} \right)}\nolimits^2 s + 4\mathop {\left({ - {{{\rm \omega} _B } \over {\rm \omega}}} \right)}\nolimits^2 \left({\sin^3 \left({{{\rm \theta} \over 2}} \right)\cos\left({{{\rm \theta} \over 2}} \right)} \right)d_s {\rm \theta} } \over \Delta }\semicolon \;$$

(16)$$d_s M = - \displaystyle{{\left({1 + d_s {\rm \theta} } \right)} \over M}d_{ss} {\rm \theta} \comma \;$$

and re-write Eq.(14) as

(17)$$\eqalign{&\displaystyle{\left( - {{\rm \omega}_G \over {\rm \omega}} \right)^2 s + 4 \left(- {{\rm \omega}_B \over {\rm \omega}} \right)^2 \left(\sin^3 \left({{\rm \theta} \over 2} \right) \cos \left({{\rm \theta} \over 2} \right) \right) d_s {\rm \theta} \over \Delta} + \Delta \displaystyle{\left(1 + d_s {\rm \theta} \right) \over M^2 \left(1 - M^2 \right)} d_{ss} {\rm \theta} \cr &= - \displaystyle{{\rm \omega}_B \over {\rm \omega}} \,\ast\, \displaystyle{\left( - {{\rm \omega}_B \over {\rm \omega}} \, \ast \, 2 \sin^2 \left({{\rm \theta} \over 2} \right) \right) \over \Delta} \displaystyle{M^2 \over \sqrt{1 - M^2}} \,\ast \, \sin {\rm \theta}.}$$

We denote the direction of electric field as $\overrightarrow{e_\parallel} $ and the direction vertical to electric field as $\overrightarrow {e_ \bot } $. Clearly, the contribution from the motion of charged particle along $\overrightarrow {e_ \bot } $ to Γ_max is not clear enough. The formula d _t Γ = E · υ reflects that the dependence of Γ on ${\rm \upsilon} _\parallel $. Because Γ depends on both ${\rm \upsilon} _\parallel $ and ${\rm \upsilon} _ \bot $, this formula reflects the relation between Γ and ${\rm \upsilon} _\parallel $ when ${\rm \upsilon} _ \bot $ is given and hence should be denoted more strictly as $d_t \Gamma \vert _{d_t {\rm \upsilon} _ \bot = 0} = E \cdot {\rm \upsilon} $. Namely, we can always obtain d _t Γ = E · υ from RNEs if $d_t {\rm \upsilon} _ \bot = 0$. Clearly, once $d_t {\rm \upsilon} _ \bot = 0$ cannot be warranted, Γ (t), which should be equal to $\Gamma \left(0 \right)+ \vint d_t \Gamma \vert _{d_t {\rm \upsilon} _ \bot = 0} dt + \vint d_t \Gamma \vert _{d_t {\rm \upsilon} _\parallel = 0} dt$, might be different from Γ ( 0 ) + ∫ E · υ dt.

The 3D RNEs is ultimately reduced to Eq. (17), a second-order differential equation of $z = \vint {\rm \upsilon} _ \bot dt$. All related quantities can be calculated from this equation. In this compound field setup, due to the DC electric field, the motion of a charged particle is no longer periodic even though the laser is a periodic stimulus. As shown in Figure 1, Γ rises with respect to t in an oscillatory manner. If no DC electric field is applied, i.e., ${{{\rm \omega} _G } \over {\rm \omega} } = 0$, Γ will periodically oscillate between 1 and a time-independent Γ_max, and Γ_max −1 is proportional to ${{{\rm \omega} _B } \over {\rm \omega} } = {{eA_{\rm max} } \over {mc^2 }}$ (Scheid & Hora, Reference Scheid and Hora1989; Goreslavsky et al., Reference Goreslavsky, Fedorov and Kilpio1995; Lin et al., Reference Lin, Liu, Du and Wang2013). If ${{{\rm \omega} _G } \over {\rm \omega} } \ll {{{\rm \omega} _B } \over {\rm \omega} }$, Γ will do nearly periodical oscillation (see green dotted line in the down panel of Fig. 1). Likewise, if no laser is applied, i.e., ${{{\rm \omega} _B } \over {\rm \omega}} = 0$, Γ will monotonically rises with respect to t (see black solid line in Fig. 2). Clearly, the oscillatory rising, which is aperiodic, reflects combined effect of the laser and the DC electric field.

Fig. 1. Example of time history of Γ.

Fig. 2. Example of time history of Γ.

It is worthy to note that under the same energy density of the stimulus, aperiodic motion favors higher Γ_max than periodic one. Because the electromagnetic energy density is ${^\sim} \left({{\rm \omega} _B \over {\rm \omega}} \right)^2 $, the energy density of a laser beam of ${{{\rm \omega} _B } \over {\rm \omega}}={\sqrt 2} a$ will be equal to the summation of those of two laser beams of ${{{\rm \omega} _B } \over {\rm \omega} }=a$. As shown in Figure 1, for nearly same energy density, $\left({{\textstyle{{{\rm \omega} _G } \over {\rm \omega} }}\comma \; {\textstyle{{{\rm \omega} _B } \over {\rm \omega} }}} \right)=\left({0.01\comma \; 0.01} \right)$ is more favorable to high Γ than $\left({{\textstyle{{{\rm \omega} _G } \over {\rm \omega} }}\comma \; {\textstyle{{{\rm \omega} _B } \over {\rm \omega} }}} \right)=\left({0.0001\comma \; 0.015} \right)$ (here because of ${\sqrt 2}=1.414 ^{\sim} 1.5$, we approximate 0.015 as $0.01\, \ast\, {\sqrt 2} $. ). More interest, high Γ_max is mainly contributed by high ${\rm \upsilon} _ \bot ^2 $ instead of ${\rm \upsilon} _\parallel ^2 $. As shown in Figure 3, d _s θ, which is just dimensionless ${\rm \upsilon} _ \bot $ subtracting 1, obviously differs from −1. This implies that ${\rm \upsilon} _ \bot $ obviously differs from 0 and hence have marked contribution to high Γ_max-value. Here, we should also note that if ${{{\rm \omega} _B } \over {\rm \omega} } = 0$, there will be always ${\rm \upsilon} _ \bot = 0$. Namely, the laser is necessary for ensuring ${\rm \upsilon} _ \bot \ne 0$.

Fig. 3. Example of time history of d _sθ.

According to strict theories (Scheid & Hora, Reference Scheid and Hora1989; Goreslavsky et al., Reference Goreslavsky, Fedorov and Kilpio1995), the laser enables particle's velocity to be periodic. The periodicity limits Γ to be $\les \Gamma_{\rm max}$ whose value is determined by laser strength E _max. Usually people choose to increase Γ_max by enhancing E _max under same setup (i.e., a single laser beam). Above results have shown that trying different setups might be more effective to find better acceleration scheme. The compound field setup described above has clearly illustrated a typical example of better acceleration. In short, how to harness the motion vertical to the electric field is a worthy issue for ensuring higher Γ_max. Because of the formula d _t Γ = E · υ, the importance of ${\rm \upsilon} _ \bot ^2 $ to higher Γ_max has been underestimated greatly.

A simple estimation can illustrate the effect of ${\rm \upsilon} _ \bot ^2 $ to Γ_max. According to Eq.(10), if ${\textstyle{{{\rm \omega} _G } \over {\rm \omega} }} = 0$ and ${\rm \upsilon} _ \bot ^2 \equiv 0$ exist, there will be $\Gamma = 1/\sqrt {1 - \mathop {\left({d_s {\textstyle{x \over {\rm \lambda} }}} \right)}\nolimits^2 } = \sqrt {1 + \mathop {\left({ - {\textstyle{{{\rm \omega} _B } \over {\rm \omega} }} \,\ast\, 2\sin^2 \left({{\textstyle{{\rm \theta} \over 2}}} \right)} \right)}\nolimits^2 } $, which can also be derived from $d_s \left[{1/\sqrt {1 - \mathop {\left({d_s {\textstyle{x \over {\rm \lambda} }}} \right)}\nolimits^2 } } \right]= {\textstyle{{{\rm \omega} _B } \over {\rm \omega} }}\sin{\rm \theta} \,\ast\, {d_s} {\textstyle{x \over {\rm \lambda} }}$. In contrast, if ${\textstyle{{{\rm \omega} _G } \over {\rm \omega} }} = 0$ exists but ${\rm \upsilon} _ \bot ^2 \equiv 0$ not, Eq.(10) will yield $\Gamma = 1/\sqrt {1 - \mathop {\left({d_s {\textstyle{x \over {\rm \lambda} }}} \right)}\nolimits^2 - \mathop {\left({d_s {\textstyle{z \over {\rm \lambda} }}} \right)}\nolimits^2 } = \sqrt {1 + \mathop {\left({ - {\textstyle{{{\rm \omega} _B } \over {\rm \omega} }} \,\ast\, 2\sin^2 \left({{\textstyle{{\rm \theta} \over 2}}} \right)} \right)}\nolimits^2 } /$${\sqrt {1 - \mathop {\left({d_s {\textstyle{z \over {\rm \lambda} }}} \right)}\nolimits^2 } \,\gt\,} \sqrt {1 + \mathop {\left({ - {\textstyle{{{\rm \omega} _B } \over {\rm \omega} }} \,\ast\, 2\sin^2 \left({{\textstyle{{\rm \theta} \over 2}}} \right)} \right)}\nolimits^2 } $. The > well illustrates the underestimation of Γ if ${\rm \upsilon} _ \bot ^2 \equiv 0$ is taken for granted.

Moreover, for ${\textstyle{{{\rm \omega} _G } \over {\rm \omega} }} = 0$, there will be $d_s {\textstyle{y \over {\rm \lambda} }} \equiv 0$ and

(18)$$\eqalign{&\displaystyle{4 \left( - {{\rm \omega}_B \over {\rm \omega}} \right)^2 \left(\sin^3 \left({{\rm \theta} \over 2} \right) \cos \left({{\rm \theta} \over 2} \right) \right)d_s {\rm \theta} \over \Delta_0} + \Delta_0 \displaystyle{\left(1 + d_s {\rm \theta} \right) \over M^2 \left(1 - M^2 \right)} d_{ss} {\rm \theta} \cr & = - \displaystyle{{\rm \omega}_B \over {\rm \omega}} \, \ast \, \displaystyle{\left( - {{\rm \omega}_B \over {\rm \omega}} \, \ast \, 2 \sin^2 \left({{\rm \theta} \over 2} \right) \right) \over \Delta_0} \displaystyle{M^2 \over \sqrt{1 - M^2}} \, \ast \, \sin {\rm \theta} \comma \; }$$

where

(19)$$\Delta _0=\sqrt {1+\mathop {\left({ - \displaystyle{{{\rm \omega} _B } \over {\rm \omega} } \,\ast\, 2\sin^2 \left({\displaystyle{{\rm \theta} \over 2}} \right)} \right)}\nolimits^2 }.$$

A straightforward deduction can confirm that Eq.(18) corresponds to a first integral and hence has periodic solutions. In contrast, Eq.(17) does not have periodic solutions.

People have been aware that the combination filed can lead to significant acceleration. For example, Yin et al., (Reference Yin, Albright, Heglich and Fernández2006) and Flippo et al., (Reference Flippo, Hegelich and Albright2007) attributed high-energy ions appearing in the interaction of intense laser with thin solid target to the acceleration by the combination of a transverse field (i.e., laser) and a longitudinal electric field (i.e., sheath field). In the cases they studied, the laser is “actively” applied and the longitudinal electric field is “passively” generated. Moreover, the longitudinal electric field is time-dependent and its direction is fixed to be along the propagation direction of the laser (and hence vertical to that of laser magnetic field). In contrast, the case we studied is of more flexible arrangement of the fields. Both the transverse field and the longitudinal one are “actively” applied, the longitudinal field is time-independent and flexible adjustment on its direction is available. Such a flexible arrangement enables us to make a full utilization of the merit of the combination acceleration.

3. SUMMARY

For higher efficiency, we extend investigation on laser direction vacuum acceleration to the cases of multiple stimuli. The advantage of multiple stimuli over single stimulus is confirmed by strict theory and numerical results. Especially, the importance of the motion vertical to the direction of electric field to the acceleration is reflected by strict analysis and numerical results. This implies that how to make a full utilization of ${\rm \upsilon} _ \bot ^2 $ is essential to warrant high-efficiency acceleration.